A Natural Identity for Exponential Families with Applications in Multiparameter Estimation

A random variable X is said to have distribution in the class E₀ if, for some real valued, positive function a (), the identity E\ (X -) g (X) \ = E\a (X) g' (X) \ holds for any absolutely continuous real valued function g () satisfying E|a (X) g' (X) | 2. Suppose X₁, , Xₚ, p 3, are independently distributed with distributions in E₀, for some function a (), and with means ₁, , ₚ. Define b (x) = a (x) ^-1 dx, where the integral is interpreted as indefinite, Bᵢ = b (Xᵢ), S = ᵖ₈=₁ Bᵢ², X' = (X₁, , Xₚ) and B' = (B₁, , Bₚ). Then the estimator X - ( (p - 2) /S) B dominates X if sum of squared error loss is assumed. Similar estimators are obtained, when p 4, which shrink towards an origin determined by the data. There are corresponding results for some discrete exponential families.